I’ve been trying to extend the results of the work described in this post, and following a suggestion of Noam Elkies have changed my search strategy from points corresponding to simple EDS triples to those given by (A,u,c) parametrisations as described here. Experimenting with these revealed some serious deficiencies with the height function in SAGE, so EDS are still involved at a practical level- but with enough magma licenses, one could just test all the points directly.
In good news for maths but perhaps bad news for my would-be paper, this straightforward approach has yielded several new (and record-breaking) examples of small height points, which I’ve added to the tables. A few also match or improve upon the best known values for most, highest, and most consecutive integral multiples. The table below summarises these: for the point [0,0] on the curve E:Y2 + a1XY + a3Y = X3 + a2X2,with P the corresponding point on the minimal model of E, we list the values of n≤50 such that nP is integral.
| w | (A,u,c) | [a1,a2,a3] | n | |
| A | √2 | (w+1,w-1,1) | [-13w – 23, 49w + 70, -1820w – 2576] | 1-10,12,13,15-20,25,35 |
| B | √6 | (w-3,w-3,1) | [-12443w + 30479, -230496005w + 564597600, -7958566915120w + 19494428025840] | 1-15,19,20,21,23,24,26,29 |
| C | √3 | (-2w-4,-w-3,1) | [17298w + 29961, 332452269w + 575824221, 9670381784073w + 16749592578603] | 1-12,14,15,18,24,29 |
| D | √3 | (1,2w-4,1) | [2856w – 4944, 42937344w – 74369664, -746077879296w + 1292244793344] | 1-12,14,15,16,18,27 |
| E | √7 | (2w-6,1-w,1) | [-5922w + 15669, -35749431w + 94584105, -543103643331w + 1436917176387] | 1-11,13,15,17,21,26 |
| F | √3 | (-2w-4,w+1,1) | [1086w + 1881, 716035w + 1240209, 1277410855w + 2212540503] | 1-8,10,11,12,14,15,16,21,22 |
| G | √5/2+1/2 | (w,w-2,1) | [4-w,6w-18,60w-90] | 1-15,18,22 |
Highest integral multiples: Over Q, the record is 31; this is exceeded by point A, at 35.
Most integral multiples: Over Q, the record is 16. All seven examples above match or exceed this: point B has the most, at 22; followed by A at 20; C,D and G at 17; and E and F at 16.
Most consecutive integral multiples: Over Q, the record is 14: points B and G both beat this, with their first 15 multiples being integral.